1. Field of the Invention
This invention relates generally to digital communications and particularly to signal modulation techniques. Still more particularly, this invention relates to phase shift keying, which is a digital modulation scheme that conveys data by changing, or modulating, the phase of a reference signal.
2. Description of the Related Art
In telecommunications, modulation is the process of varying a periodic waveform in order to use that signal to convey a message or to transmit information. Any digital modulation scheme uses a finite number of distinct signals to represent digital data. Phase shift keying (PSK) is a digital modulation scheme that conveys data by changing, or modulating, the phase of a reference signal (the carrier wave). PSK uses a finite number of phases, each assigned a unique pattern of binary bits. Usually each phase encodes an equal number of bits. Each pattern of bits forms the symbol that is represented by the particular phase. In digital communications, a symbol is the smallest unit of data transmitted at one time.
Traditional digital modulation techniques include binary phase shift keying and quadrature phase shift keying. Binary phase shift keying (BPSK) has only one basis function, which is expressed as
                                          ϕ            ⁡                          (              t              )                                =                                                    2                T                                      ⁢                          cos              ⁡                              (                                  2                  ⁢                  π                  ⁢                                                                          ⁢                                      f                    c                                    ⁢                  t                                )                                                    ,                            (        1        )            
where fc is the carrier wave frequency and T represents the symbol duration and the basis function is defined only for times between 0 and T. The BPSK waveforms, s1(t)=−s2(t), are represented ass1(t)=√{square root over (Eb)}φ(t)=−s2(t),  (2)where Eb is the energy per bit.
In the presence of zero mean additive white Gaussian noise (AWGN) with a variance of No/2, the probability of bit error using maximum likelihood (ML) detection is
                                          P            b                    =                      Q            ⁡                          (                                                                    2                    ⁢                                          E                      b                                                                            N                    o                                                              )                                      ,                            (        3        )            where Eb is the energy per bit, and the Q function is
                              Q          ⁡                      (            x            )                          =                              1                                          2                ⁢                π                                              ⁢                                    ∫              0              ∞                        ⁢                                          e                                                      -                                          x                      2                                                        2                                            ⁢                                                          ⁢                                                ⅆ                  x                                .                                                                        (        4        )            
M-ary phase shift keying (MPSK) uses two orthonormal basis functions
                                          ϕ            1                    ⁡                      (            t            )                          =                              A                                          E                s                                              ⁢                      cos            ⁡                          (                              2                ⁢                π                ⁢                                                                  ⁢                                  f                  c                                ⁢                t                            )                                                          (        5        )            and
                                                        ϕ              2                        ⁡                          (              t              )                                =                                    -                              A                                                      E                    1                                                                        ⁢                          sin              ⁡                              (                                  2                  ⁢                  π                  ⁢                                                                          ⁢                                      f                    c                                    ⁢                  t                                )                                                    ,                            (        6        )            where A is the amplitude, fc=m/T with m being a positive integer and the basis functions are defined only for time values between 0 and T. The constraint on the frequency allows the two basis functions to be orthogonal to each other.
Additionally, Es represents the energy per symbol and equals A2T/2. The signals sn(t), where n=1, 2, . . . , M are written as
                                          s            n                    ⁡                      (            t            )                          =                                            E              s                                ⁢                                    ⌊                                                cos                  ⁡                                      (                                                                                            π                          M                                                ⁢                                                  (                                                                                    2                              ⁢                              n                                                        -                            1                                                    )                                                ⁢                                                                              ϕ                            1                                                    ⁡                                                      (                            t                            )                                                                                              +                                              sin                        ⁡                                                  (                                                                                    π                              M                                                        ⁢                                                          (                                                                                                2                                  ⁢                                  n                                                                -                                1                                                            )                                                                                )                                                                                      )                                                  ⁢                                                      ϕ                    2                                    ⁡                                      (                    t                    )                                                              ⌋                        .                                              (        7        )            
For quadrature phase shift keying (QPSK) where M=4, the probability of bit error using ML detection in AWGN is the same as is the same as in Equation (3) because QPSK in effect is two independent BPSK channels. Note that even though the bit error rates are identical, QPSK has twice the bandwidth efficiency of BPSK.
The bit error probability for M-PSK in general is
                              P          b                =                              2                                          log                2                            ⁢              M                                ⁢                                    Q              ⁡                              (                                                      sin                    ⁡                                          (                                              π                        M                                            )                                                        ⁢                                                                                    log                        2                                            ⁢                      M                      ⁢                                                                        2                          ⁢                                                      E                            b                                                                                                    N                          o                                                                                                                    )                                      .                                              (        8        )            Equation (8) is accurate when gray coding is used to code adjacent symbols.
Another commonly used modulation is M-quadrature modulation (QAM), which is simply amplitude modulation over two quadrature channels. The bit error probability for M-QAM in general is
                              P          b                =                                            4              ⁢                              (                                  1                  -                                      1                                          M                                                                      )                                                                    log                2                            ⁢              M                                ⁢                                    Q              ⁡                              (                                                                            3                      ⁢                                              log                        2                                            ⁢                                              ME                        b                                                                                                            (                                                  M                          -                          1                                                )                                            ⁢                                              N                        o                                                                                            )                                      .                                              (        9        )            
In the simplest modulation schemes such as binary phase-shift keying, only one bit of data (i.e., a 0 or 1) is transmitted at a time depending on the phase of the transmitted signal. However, in a more complex scheme such as 16-QAM, four bits of data are transmitted simultaneously, resulting in a symbol rate (or baud rate) that is equal to one quarter of the bit rate.
Owing to PSK's simplicity, it is widely used in existing technologies. The most popular wireless local area network (LAN) wireless standard, IEEE 802.11b uses a variety of different PSK techniques depending on the data-rate required. At the basic-rate of 1 Mbit/s, it uses differential BPSK. To provide the extended-rate of 2 Mbit/s, DQPSK is used. In reaching 5.5 Mbit/s and the full-rate of 11 Mbit/s, QPSK is employed coupled with complementary code keying. The higher-speed wireless LAN standard, IEEE 802.11g has eight data rates: 6, 9, 12, 18, 24, 36, 48 and 54 Mbit/s. The 6 and 9 Mbit/s modes use BPSK. The 12 and 18 Mbit/s modes use QPSK. The fastest four modes use forms of quadrature amplitude modulation.
Because of its simplicity BPSK is appropriate for low-cost passive transmitters, and is used in radio frequency identification (RFID) standards such as International Organization for Standardization (ISO) 14443 which has been adopted for biometric passports, credit cards such as American Express's Express Pay and many other applications.